Theorem redcwlpo 16952

Description: Decidability of real number equality implies the Weak Limited Principle of Omniscience (WLPO). We expect that we'd need some form of countable choice to prove the converse.

Here's the outline of the proof. Given an infinite sequence F of zeroes and ones, we need to show the sequence is all ones or it is not. Construct a real number A whose representation in base two consists of a zero, a decimal point, and then the numbers of the sequence. This real number will equal one if and only if the sequence is all ones (redcwlpolemeq1 16951). Therefore decidability of real number equality would imply decidability of whether the sequence is all ones.

Because of this theorem, decidability of real number equality is sometimes called "analytic WLPO".

WLPO is known to not be provable in IZF (and most constructive foundations), so this theorem establishes that we will be unable to prove an analogue to qdceq 10628 for real numbers. (Contributed by Jim Kingdon, 20-Jun-2024.)

Assertion

Ref	Expression
redcwlpo	|- ( A. x e. RR A. y e. RR DECID x = y -> om e. WOmni )

Distinct variable group:   x, y

Proof of Theorem redcwlpo

Dummy variables f i j z are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		simpl 109	. . . . . 6 |- ( ( A. x e. RR A. y e. RR DECID x = y /\ f e. ( { 0 , 1 } ^m NN ) ) -> A. x e. RR A. y e. RR DECID x = y )
2		elmapi 6917	. . . . . . . . 9 |- ( f e. ( { 0 , 1 } ^m NN ) -> f : NN --> { 0 , 1 } )
3	2	adantl 277	. . . . . . . 8 |- ( ( A. x e. RR A. y e. RR DECID x = y /\ f e. ( { 0 , 1 } ^m NN ) ) -> f : NN --> { 0 , 1 } )
4		oveq2 6066	. . . . . . . . . . 11 |- ( i = j -> ( 2 ^ i ) = ( 2 ^ j ) )
5	4	oveq2d 6074	. . . . . . . . . 10 |- ( i = j -> ( 1 / ( 2 ^ i ) ) = ( 1 / ( 2 ^ j ) ) )
6		fveq2 5675	. . . . . . . . . 10 |- ( i = j -> ( f ` i ) = ( f ` j ) )
7	5, 6	oveq12d 6076	. . . . . . . . 9 |- ( i = j -> ( ( 1 / ( 2 ^ i ) ) x. ( f ` i ) ) = ( ( 1 / ( 2 ^ j ) ) x. ( f ` j ) ) )
8	7	cbvsumv 12071	. . . . . . . 8 |- sum_ i e. NN ( ( 1 / ( 2 ^ i ) ) x. ( f ` i ) ) = sum_ j e. NN ( ( 1 / ( 2 ^ j ) ) x. ( f ` j ) )
9	3, 8	trilpolemcl 16933	. . . . . . 7 |- ( ( A. x e. RR A. y e. RR DECID x = y /\ f e. ( { 0 , 1 } ^m NN ) ) -> sum_ i e. NN ( ( 1 / ( 2 ^ i ) ) x. ( f ` i ) ) e. RR )
10		1red 8305	. . . . . . 7 |- ( ( A. x e. RR A. y e. RR DECID x = y /\ f e. ( { 0 , 1 } ^m NN ) ) -> 1 e. RR )
11		eqeq1 2241	. . . . . . . . 9 |- ( x = sum_ i e. NN ( ( 1 / ( 2 ^ i ) ) x. ( f ` i ) ) -> ( x = y <-> sum_ i e. NN ( ( 1 / ( 2 ^ i ) ) x. ( f ` i ) ) = y ) )
12	11	dcbid 846	. . . . . . . 8 |- ( x = sum_ i e. NN ( ( 1 / ( 2 ^ i ) ) x. ( f ` i ) ) -> (DECID x = y <-> DECID sum_ i e. NN ( ( 1 / ( 2 ^ i ) ) x. ( f ` i ) ) = y ) )
13		eqeq2 2244	. . . . . . . . 9 |- ( y = 1 -> ( sum_ i e. NN ( ( 1 / ( 2 ^ i ) ) x. ( f ` i ) ) = y <-> sum_ i e. NN ( ( 1 / ( 2 ^ i ) ) x. ( f ` i ) ) = 1 ) )
14	13	dcbid 846	. . . . . . . 8 |- ( y = 1 -> (DECID sum_ i e. NN ( ( 1 / ( 2 ^ i ) ) x. ( f ` i ) ) = y <-> DECID sum_ i e. NN ( ( 1 / ( 2 ^ i ) ) x. ( f ` i ) ) = 1 ) )
15	12, 14	rspc2v 2937	. . . . . . 7 |- ( ( sum_ i e. NN ( ( 1 / ( 2 ^ i ) ) x. ( f ` i ) ) e. RR /\ 1 e. RR ) -> ( A. x e. RR A. y e. RR DECID x = y -> DECID sum_ i e. NN ( ( 1 / ( 2 ^ i ) ) x. ( f ` i ) ) = 1 ) )
16	9, 10, 15	syl2anc 411	. . . . . 6 |- ( ( A. x e. RR A. y e. RR DECID x = y /\ f e. ( { 0 , 1 } ^m NN ) ) -> ( A. x e. RR A. y e. RR DECID x = y -> DECID sum_ i e. NN ( ( 1 / ( 2 ^ i ) ) x. ( f ` i ) ) = 1 ) )
17	1, 16	mpd 13	. . . . 5 |- ( ( A. x e. RR A. y e. RR DECID x = y /\ f e. ( { 0 , 1 } ^m NN ) ) -> DECID sum_ i e. NN ( ( 1 / ( 2 ^ i ) ) x. ( f ` i ) ) = 1 )
18	3, 8	redcwlpolemeq1 16951	. . . . . 6 |- ( ( A. x e. RR A. y e. RR DECID x = y /\ f e. ( { 0 , 1 } ^m NN ) ) -> ( sum_ i e. NN ( ( 1 / ( 2 ^ i ) ) x. ( f ` i ) ) = 1 <-> A. z e. NN ( f ` z ) = 1 ) )
19	18	dcbid 846	. . . . 5 |- ( ( A. x e. RR A. y e. RR DECID x = y /\ f e. ( { 0 , 1 } ^m NN ) ) -> (DECID sum_ i e. NN ( ( 1 / ( 2 ^ i ) ) x. ( f ` i ) ) = 1 <-> DECID A. z e. NN ( f ` z ) = 1 ) )
20	17, 19	mpbid 147	. . . 4 |- ( ( A. x e. RR A. y e. RR DECID x = y /\ f e. ( { 0 , 1 } ^m NN ) ) -> DECID A. z e. NN ( f ` z ) = 1 )
21	20	ralrimiva 2617	. . 3 |- ( A. x e. RR A. y e. RR DECID x = y -> A. f e. ( { 0 , 1 } ^m NN )DECID A. z e. NN ( f ` z ) = 1 )
22		nnex 9260	. . . 4 |- NN e. _V
23		iswomninn 16947	. . . 4 |- ( NN e. _V -> ( NN e. WOmni <-> A. f e. ( { 0 , 1 } ^m NN )DECID A. z e. NN ( f ` z ) = 1 ) )
24	22, 23	ax-mp 5	. . 3 |- ( NN e. WOmni <-> A. f e. ( { 0 , 1 } ^m NN )DECID A. z e. NN ( f ` z ) = 1 )
25	21, 24	sylibr 134	. 2 |- ( A. x e. RR A. y e. RR DECID x = y -> NN e. WOmni )
26		nnenom 10820	. . 3 |- NN ~~ om
27		enwomni 7474	. . 3 |- ( NN ~~ om -> ( NN e. WOmni <-> om e. WOmni ) )
28	26, 27	ax-mp 5	. 2 |- ( NN e. WOmni <-> om e. WOmni )
29	25, 28	sylib 122	1 |- ( A. x e. RR A. y e. RR DECID x = y -> om e. WOmni )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105  DECID wdc 842   = wceq 1398   e. wcel 2205   A.wral 2522   _Vcvv 2815   {cpr 3695   class class class wbr 4114   omcom 4717   -->wf 5353   ` cfv 5357  (class class class)co 6058   ^m cmap 6895   ~~ cen 6986  WOmnicwomni 7467   RRcr 8142   0cc0 8143   1c1 8144   x. cmul 8148   / cdiv 8963   NNcn 9254   2c2 9305   ^cexp 10924   sum_csu 12063

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2207  ax-14 2208  ax-ext 2216  ax-coll 4230  ax-sep 4233  ax-nul 4241  ax-pow 4292  ax-pr 4327  ax-un 4559  ax-setind 4664  ax-iinf 4715  ax-cnex 8234  ax-resscn 8235  ax-1cn 8236  ax-1re 8237  ax-icn 8238  ax-addcl 8239  ax-addrcl 8240  ax-mulcl 8241  ax-mulrcl 8242  ax-addcom 8243  ax-mulcom 8244  ax-addass 8245  ax-mulass 8246  ax-distr 8247  ax-i2m1 8248  ax-0lt1 8249  ax-1rid 8250  ax-0id 8251  ax-rnegex 8252  ax-precex 8253  ax-cnre 8254  ax-pre-ltirr 8255  ax-pre-ltwlin 8256  ax-pre-lttrn 8257  ax-pre-apti 8258  ax-pre-ltadd 8259  ax-pre-mulgt0 8260  ax-pre-mulext 8261  ax-arch 8262  ax-caucvg 8263

This theorem depends on definitions:  df-bi 117  df-dc 843  df-3or 1006  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ne 2415  df-nel 2510  df-ral 2527  df-rex 2528  df-reu 2529  df-rmo 2530  df-rab 2531  df-v 2817  df-sbc 3046  df-csb 3142  df-dif 3216  df-un 3218  df-in 3220  df-ss 3227  df-nul 3513  df-if 3625  df-pw 3676  df-sn 3700  df-pr 3701  df-op 3703  df-uni 3920  df-int 3955  df-iun 3998  df-br 4115  df-opab 4177  df-mpt 4178  df-tr 4214  df-id 4419  df-po 4422  df-iso 4423  df-iord 4492  df-on 4494  df-ilim 4495  df-suc 4497  df-iom 4718  df-xp 4760  df-rel 4761  df-cnv 4762  df-co 4763  df-dm 4764  df-rn 4765  df-res 4766  df-ima 4767  df-iota 5317  df-fun 5359  df-fn 5360  df-f 5361  df-f1 5362  df-fo 5363  df-f1o 5364  df-fv 5365  df-isom 5366  df-riota 6011  df-ov 6061  df-oprab 6062  df-mpo 6063  df-1st 6347  df-2nd 6348  df-recs 6549  df-irdg 6614  df-frec 6635  df-1o 6660  df-2o 6661  df-oadd 6664  df-er 6780  df-map 6897  df-en 6989  df-dom 6990  df-fin 6991  df-womni 7468  df-pnf 8326  df-mnf 8327  df-xr 8328  df-ltxr 8329  df-le 8330  df-sub 8462  df-neg 8463  df-reap 8866  df-ap 8873  df-div 8964  df-inn 9255  df-2 9313  df-3 9314  df-4 9315  df-n0 9514  df-z 9595  df-uz 9872  df-q 9970  df-rp 10005  df-ico 10246  df-fz 10362  df-fzo 10499  df-seqfrec 10834  df-exp 10925  df-ihash 11164  df-cj 11552  df-re 11553  df-im 11554  df-rsqrt 11708  df-abs 11709  df-clim 11989  df-sumdc 12064

This theorem is referenced by: (None)